Theorem necon2i 2564

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Hypothesis

Ref	Expression
necon2i.1	⊢ (A = B → C ≠ D)

Assertion

Ref	Expression
necon2i	⊢ (C = D → A ≠ B)

Proof of Theorem necon2i

Step	Hyp	Ref	Expression
1		necon2i.1	. . 3 ⊢ (A = B → C ≠ D)
2	1	neneqd 2533	. 2 ⊢ (A = B → ¬ C = D)
3	2	necon2ai 2562	1 ⊢ (C = D → A ≠ B)

This proof depends on syntax axioms:   → wi 4   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by:  map0  6026